MATH · LAW OF SINES
Law of Sines Calculator
Solve triangles using the law of sines. Supports ASA, AAS, and SSA modes. The SSA ambiguous case shows 0, 1, or 2 valid solutions automatically.
About This Calculator
The law of sines states that in any triangle, the ratio of a side to the sine of its opposite angle is constant: a/sin A = b/sin B = c/sin C. This calculator solves ASA (two angles + included side), AAS (two angles + non-included side), and the SSA ambiguous case — which can produce zero, one, or two valid triangles.
How It Works
Choose the mode (ASA, AAS, or SSA) and enter the known values. For ASA and AAS the solution is always unique (one triangle, or an error if the angles sum to 180° or more). For SSA, sin B = b·sin A / a is computed: if sin B > 1 there is no valid triangle; if sin B = 1 there is exactly one right triangle; otherwise up to two triangles are shown (the acute B solution and the obtuse B solution, if both produce a positive third angle).
The Formula
a / sin A = b / sin B = c / sin C
- a, b, c
- side lengths
- A, B, C
- angles (degrees) opposite the corresponding sides
Frequently Asked Questions
- What is the SSA ambiguous case?
- In SSA (two sides and an angle opposite one of them), the given measurements may correspond to two different triangles, exactly one, or none. This happens because the inverse sine function (arcsin) can return two angles — an acute and an obtuse value — both of which might form a valid triangle with the given side.
- When does SSA have two solutions?
- Two solutions exist when angle A is acute and sin B < 1, AND the obtuse solution B₂ = 180° − B₁ still satisfies A + B₂ < 180°. If A ≥ 90° or the obtuse solution would exceed 180°, only one solution (or none) is found.
- How is AAS different from ASA?
- In ASA the known side is between the two known angles (side c is opposite the unknown angle C). In AAS the known side is opposite one of the known angles (side a is opposite angle A). The calculation differs in which ratio you start with.
- What if my angles sum to 180° or more?
- A valid triangle requires the three interior angles to sum to exactly 180°. If two given angles already sum to 180° or more, no triangle is possible and the calculator returns an error.